Differential equations are the foundation of mathematical modeling—the tools we use to describe how things change. Whether it’s the motion of planets, the spread of a virus, or the flow of electricity, these equations give us a way to connect what we know about a system to how it will evolve over time or space. This chapter serves as your starting point for understanding why differential equations matter and how they’re built.
We’ll begin with the essential question: “What exactly is a differential equation?” From there, we’ll break the concept into manageable pieces—learning the vocabulary, identifying key components like terms and coefficients, and understanding how equations are classified by order and linearity. These ideas form the language that the rest of the book will speak.
By the end of this chapter, you’ll be able to look at an unfamiliar equation and recognize what makes it a differential equation, describe its structure, and understand the basic categories it might belong to. This foundation will support every method and concept that comes next.
